Riemann Curvature Tensor

The Riemann curvature tensor is a tensor field describing the intrinsic curvature of a smooth manifold equipped with a connection. In differential geometry, it usually means the curvature of the Levi-Civita connection determined by a metric tensor. It measures how covariant differentiation fails to commute and provides the local curvature information underlying sectional, Ricci, and scalar curvature. In general relativity, it describes the curvature of spacetime and the relative acceleration of freely falling bodies. (damtp.cam.ac.uk)

Definition and conventions

Let $\nabla$ be a connection on the tangent bundle of $M$ . For smooth vector fields $X,Y,Z$ , one common convention defines

R(X,Y)Z =\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ -\nabla_{[X,Y]}Z,

where $[X,Y]$ is their Lie bracket. The subtraction involving this bracket makes the expression tensorial: its value at a point depends only on the vectors there, rather than on their extensions to neighboring points. (damtp.cam.ac.uk)

At each point, $R$ is a multilinear map from three copies of the tangent space to that tangent space. It is consequently a tensor of type $(1,3)$ . Using the metric to lower its output index produces a type- $(0,4)$ tensor. Some authors reverse the overall sign of $R$ or arrange its indices differently; formulas must therefore be compared together with their defining conventions. (math.ru.nl)

For a metric connection, the covariant derivative preserves the metric. The Levi-Civita connection also has zero torsion. These two conditions give its curvature additional symmetries that need not hold for an arbitrary connection. (web.math.ucsb.edu)

Coordinate expression

In coordinates $x^\mu$ , define components by

R(\partial_\mu,\partial_\nu)\partial_\rho =R^\sigma{}_{\rho\mu\nu}\partial_\sigma.

If $\Gamma^\sigma_{\mu\rho}$ are the Christoffel symbols, then

R^\sigma{}_{\rho\mu\nu} =\partial_\mu\Gamma^\sigma_{\nu\rho} -\partial_\nu\Gamma^\sigma_{\mu\rho} +\Gamma^\sigma_{\mu\lambda}\Gamma^\lambda_{\nu\rho} -\Gamma^\sigma_{\nu\lambda}\Gamma^\lambda_{\mu\rho}.

Here $\partial_\mu$ denotes a partial derivative, and repeated indices follow the Einstein summation convention. Although connection coefficients are not tensor components, this combination transforms tensorially. (damtp.cam.ac.uk)

For the Levi-Civita connection,

\Gamma^\sigma_{\mu\nu} =\frac12g^{\sigma\lambda} (\partial_\mu g_{\lambda\nu} +\partial_\nu g_{\lambda\mu} -\partial_\lambda g_{\mu\nu}),

where $g^{\sigma\lambda}$ is the inverse metric. Thus curvature involves second derivatives of the metric and products of its first derivatives. Normal coordinates make the Christoffel symbols vanish at a chosen point, but do not generally make their derivatives—or curvature—vanish there. Nonzero curvature cannot be removed by a coordinate change. (davidtong.org)

Symmetries and Bianchi identities

Write $R_{abcd}=g_{ae}R^e{}_{bcd}$ . For Levi-Civita curvature,

R_{abcd}=-R_{bacd},\qquad R_{abcd}=-R_{abdc},\qquad R_{abcd}=R_{cdab}.

These identities express antisymmetry within each index pair and symmetry under interchange of the pairs. The first, or algebraic, Bianchi identity is

R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0.

The second, or differential, identity is

(\nabla_XR)(Y,Z)+(\nabla_YR)(Z,X) +(\nabla_ZR)(X,Y)=0.

Together, the algebraic identities reduce the number of independent components in dimension $n$ to

\frac{n^2(n^2-1)}{12}.

There is one independent component in dimension two, six in dimension three, and twenty in dimension four. These are pointwise algebraic counts, not counts of dynamical degrees of freedom. (web.math.ucsb.edu)

Geometric meaning

Parallel transport compares vectors along a curve using the connection. Transport around a sufficiently small closed loop generally changes a vector; the leading change is governed by curvature and the loop’s oriented area. This connects the tensor with local holonomy, the transformations produced by transport around loops. (davidtong.org)

Curvature also controls the separation of neighboring geodesics. If $T$ is the tangent to an affinely parameterized geodesic and $J$ is a variation field arising from neighboring geodesics, then the Jacobi equation is

\frac{D^2J}{dt^2}+R(J,T)T=0.

It expresses the geometrical mechanism behind convergence and divergence of geodesics. (web.math.ucsb.edu)

This curvature is intrinsic, rather than simply a measure of bending in an ambient space. For an embedded submanifold, the Gauss equation relates intrinsic curvature to ambient curvature and the second fundamental form. They are distinct quantities: intrinsic curvature can be defined without any embedding. (cis.upenn.edu)

Sectional curvature and contractions

For linearly independent tangent vectors $u,v$ on a Riemannian manifold, sectional curvature is

K(u,v)= \frac{g(R(u,v)v,u)} {g(u,u)g(v,v)-g(u,v)^2}.

It depends only on their two-dimensional plane. Knowing sectional curvature for every plane determines the full Riemann tensor. In dimension two it equals Gaussian curvature. For constant sectional curvature $k$ ,

R(X,Y)Z=k\bigl(g(Y,Z)X-g(X,Z)Y\bigr).

Euclidean space has $k=0$ , a round sphere of radius $r$ has $k=1/r^2$ , and standard hyperbolic space has $k=-1$ . (wim.uni-mannheim.de)

Contraction produces the Ricci tensor and scalar curvature:

\operatorname{Ric}_{bd}=R^a{}_{bad}, \qquad S=g^{bd}\operatorname{Ric}_{bd}.

In dimension three, Ricci curvature determines the full tensor. In dimensions four and higher, additional information resides in the trace-free Weyl tensor. (math.ru.nl)

Role in gravitation

In relativity, geodesic deviation represents tidal gravitational effects. The Einstein field equations use Ricci and scalar curvature through

G_{\mu\nu} =\operatorname{Ric}_{\mu\nu}-\tfrac12Sg_{\mu\nu}.

The contracted Bianchi identity gives $\nabla^\mu G_{\mu\nu}=0$ , consistent with covariant conservation of stress–energy. Vacuum equations with zero cosmological constant require vanishing Ricci curvature, not vanishing Riemann curvature: vacuum spacetime can still possess tidal curvature and gravitational radiation. (davidtong.org)